Let be a space and
a normal subgroup
that is perfect (i.e. ). The plus construction is the
essentially unique space with fundamental group
and equipped with a map
which induces an
isomorphism on homology for all coefficients. It is useful in
situations where one is given a map
which is an
isomorphism on homology groups but acts wildly on homotopy groups; in
some situations, applying the plus construction can replace this
homology equivalence with a homotopy equivalence--this is how the
plus construction plays a role in group completion.